Cyclic numbers: the hidden symmetry inside 1/7

0:00Divide one by seven and you get nought point one four two eight five seven, and then the same six digits again, for ever. One, four, two, eight, five, seven. Six digits, and six is one less than seven. Hold on to that. Most repeating blocks are not worth a second look. This one is, because of what happens when you multiply it. Not the fraction. The block itself, treated as an ordinary six-figure number.

0:33One hundred and forty two thousand, eight hundred and fifty seven, doubled. Two hundred and eighty five thousand, seven hundred and fourteen. Two, eight, five, seven, one, four. Look at what is there. A two, an eight, a five, a seven, a one and a four. The same six digits you started with. Not similar digits. The same ones. Multiplication does not normally do that. Double almost any six-figure number and you get six figures with no relation to the ones you began with.

1:12So try the rest. Three times the block is four two eight five seven one. Four times is five seven one four two eight. Five times is seven one four two eight five. Six times is eight five seven one four two. Five multiplications, and not one of them introduced a digit that was not already there, or lost one that was. Every single answer is the same six digits in a different order.

1:44And not just any different order. Follow the one down the stack: it never lands in the same column twice. Here is the shape they are sliding along. Take the six digits and close them into a ring. One, four, two, eight, five, seven, and then round to the one again. Now every one of those products is this ring, read once round, starting somewhere different. Start at the one and read round: one four two eight five seven. Start at the two: two eight five seven one four, which is the double. Start at the four: four two eight five seven one, which is three times.

2:29That is what a rotation means. The digits keep their order completely. Only the starting point moves. And the six products start at six different points, one at each digit, using every position exactly once. Now the question worth asking. Why should multiplying a number rotate it? The answer is that these products are not really new numbers at all. Divide two by seven and you get nought point two eight five seven one four. That is the double, with a point in front of it.

3:07Three sevenths is nought point four two eight five seven one. Four sevenths, five sevenths, six sevenths. All five of the other products are the expansions of the other sevenths. So the six products were never six answers to a multiplication question. They were six fractions, all with a seven underneath, and the question becomes: why do six different fractions share one string of digits? For that, look at what the long division is actually doing. Not the digits it writes down, but the remainders it leaves behind.

3:48Divide one by seven and the remainders run one, three, two, six, four, five, and then one again. Six values, and they are every non-zero remainder a division by seven can possibly have. So there is one ring of remainders and it uses everything. There is nowhere else for a division by seven to be. Which means that when you divide two by seven, you are not starting a new journey. You are joining the same ring, at the point where the two is sitting. And once you are on it, every step afterwards is forced.

4:29That gives something you can check rather than believe. The six products start at positions nought, two, one, four, five and three round the ring of digits. And the six numerators sit at positions nought, two, one, four, five and three round the ring of remainders. The same list, in the same order. The digits rotate by exactly as much as the division has started further round its ring. The symmetry in the digits is not a fact about the digits at all. It is the symmetry of the division, showing through.

5:08Which tells you what a divisor has to do to behave this way. The whole argument needed one ring that everybody is on. If the remainders split into two separate rings, then fractions on the first ring cannot possibly share digits with fractions on the second. And there is a way to see, from the block alone, whether that has happened. If the block is as long as it could possibly be, one less than the divisor, then the ring has used up every available remainder and there is no room for a second one.

5:45Seven's block is six digits, and six is one less than seven. It just fits. So try thirteen, which is prime, and which looks like it ought to do the same thing. One thirteenth is nought point nought seven six nine two three. Six digits. But one less than thirteen is twelve, and six is not twelve. The block falls short. And you can watch the consequence. Three thirteenths is nought point two three nought seven six nine, which is a rotation. Four thirteenths is a rotation too. So far so good.

6:27Two thirteenths is nought point one five three eight four six. That is not a rotation of the first block. Not a single arrangement of those six digits gives it. Two thirteenths is on a different ring. And that is exactly what has happened. The twelve numerators split into two rings of six. One, three, four, nine, ten and twelve travel together. Two, five, six, seven, eight and eleven travel together, on a ring that never touches the first.

7:04Between them they use every remainder from one to twelve, exactly once each. But as two rings, not one. So thirteen gives you two separate families of digits, and no amount of multiplying will carry you from one to the other. Being prime was never the condition. The condition is the length. So the condition is worth testing properly, and in both directions. Every divisor below two hundred was taken. A hundred and eighty one of them give a repeating decimal at all. Seventeen of those have a block one less than the divisor: seven, seventeen, nineteen, twenty three, twenty nine, and on up to a hundred and ninety three.

7:53For all seventeen, multiplying the block by every single multiplier rotates it. For the other hundred and sixty four, it does not. Not a single one of them slips through in either direction. Which makes the length a genuine test rather than a hopeful description. Seventeen is the next one after seven, and its block runs sixteen digits, beginning with a nought that you must not throw away. There is one last thing to try, and it is the one that explains the edges.

8:30Multiply the block by seven. You get nine nine nine nine nine nine. Not a rotation. Nothing like one. But of course. The rotations came from the fractions, and seven sevenths is not a fraction of that kind. It is one. The ring has exactly six places on it, so there are exactly six rotations, and asking for a seventh is asking for a numerator the ring does not have.

9:01The pattern does not stop at six because it gets tired. It stops because there were only ever six fractions to go round. Which is the real lesson. What looked like a curiosity about a particular six-digit number turned out to be a fact about long division, wearing a disguise.